Angle Bisector Calculator
Calculate angle bisector lengths and division ratios
Enter Triangle Sides
Angle Bisector Theorem
The angle bisector from A divides the opposite side in the ratio of the adjacent sides:
BD/DC = AB/AC
Triangle with Angle Bisectors
Understanding Angle Bisectors
An angle bisector of a triangle is a line segment that divides an angle into two equal parts. Every triangle has three angle bisectors, and they all meet at a single point called the incenter.
Angle Bisector Length Formula
The length of the angle bisector from vertex A to side a is:
Or equivalently:
Angle Bisector Theorem
The angle bisector from a vertex divides the opposite side in the ratio of the two adjacent sides. If the bisector from A meets side BC at point D:
The Incenter
The three angle bisectors of a triangle meet at a point called the incenter (I). This point is equidistant from all three sides and is the center of the inscribed circle (incircle).
Frequently Asked Questions
What is an angle bisector?
An angle bisector is a line that divides an angle into two equal parts. In a triangle, the angle bisector from a vertex goes to the opposite side.
What is the angle bisector theorem?
The angle bisector from vertex A divides the opposite side BC in the same ratio as the adjacent sides: BD/DC = AB/AC.
Where do all three angle bisectors meet?
All three angle bisectors meet at the incenter, which is the center of the inscribed circle (incircle). It's equidistant from all three sides.
Is the incenter always inside the triangle?
Yes! The incenter is always inside the triangle, regardless of whether the triangle is acute, right, or obtuse.
How is angle bisector different from median?
An angle bisector divides an angle in half and goes to the opposite side. A median goes from a vertex to the midpoint of the opposite side. They're only the same in isosceles triangles for the angle between equal sides.
What's the relationship between angle bisector and incircle?
The incircle is the largest circle that fits inside the triangle, touching all three sides. Its center (incenter) is where all three angle bisectors meet.
Can I find the division point using the theorem?
Yes! If the bisector from A divides BC in ratio m:n, the division point D divides BC so that BD = (m/(m+n)) × BC.
What is an external angle bisector?
An external angle bisector bisects the exterior angle. It's perpendicular to the internal angle bisector and relates to the excircles.